Perturbations of the scattering resonances of an open cavity by small particles. Part II: The transverse electric polarization case

This paper is concerned with the scattering resonances of open cavities. It is a follow-up of"Perturbation of the scattering resonances of an open cavity by small particles. Part I"where the transverse magnetic polarization was assumed. In that case, using the method of matched asymptotic expansions, the leading-order term in the shifts of scattering resonances due to the presence of small particles of arbitrary shapes was derived and the effect of radiation on the perturbations of open cavity modes was characterized. The derivations were formal. In this paper, we consider the transverse electric polarization and prove a small-volume formula for the shifts in the scattering resonances of a radiating dielectric cavity perturbed by small particles. We show a strong enhancement in the frequency shift in the case of plasmonic particles. We also consider exceptional scattering resonances and perform small-volume asymptotic analysis near them. Our method in this paper relies on pole-pencil decompositions of volume integral operators.


Introduction
In this paper, which is a follow-up of [1], we consider dielectric radiating cavities [10,12,17] and rigorously obtain asymptotic formulas for the shifts in the scattering resonances that are due to a small particle of arbitrary shape. Our formula shows that the perturbations of the scattering resonances can be expressed in terms of the polarization tensor of the small particle. The scattering resonances can be degenerate or even exceptional and the small particle can be plasmonic. Our method is based on pole-pencil decompositions (see, for instance, [3,5]) of the volume integral operator associated with the radiating dielectric cavity problem. The new technique introduced in this paper can not be easily extended to the transverse magnetic case considered in [1] due to the hyper-singular character of the associated volume-integral operator.
The paper is organized as follows. In Section 2, we characterize the scattering resonances of dielectric cavities in terms of the spectrum of a volume integral operator. In Section 3, using the method of pole-pencil decompositions, we derive the leading-order term in the shifts of scattering resonances of an open dielectric cavity due to internal particles. In Section 4, using a Lippmann-Schwinger representation formula for the Green's function associated with the open cavity, we generalize the formula obtained in Section 3 to the case of external particles. In Section 5, we consider the perturbation of an open dielectric cavity by plasmonic nanoparticles. The formula obtained for the shifting of the frequencies shows a strong enhancement in the frequency shift in the case of plasmonic nanoparticles. In Section 6, we perform an asymptotic analysis for the shift of exceptional scattering resonances. The paper ends with some concluding remarks.
2 Scattering resonances of a dielectric cavity

Model
We consider the scattering of linearly polarized light by a dielectric cavity in a time-harmonic regime. Let Ω be a bounded domain in R d for d = 2, 3, with smooth boundary ∂Ω. Assume ε ≡ τ ε c + ε m inside Ω and ε = ε m outside Ω, and µ = µ m everywhere. Here, ε c , ε m , and τ are positive constants. Since we are interested in scattering resonances, we look for solutions u of the homogeneous Helmholtz equation at frequency ω: in R d , u satisfies the outgoing radiation condition. ( Let Γ m be the outgoing fundamental solution of ∆ + ε m µ m ω 2 in free space, and let G be the outgoing fundamental solution of ∆ + εµ m ω 2 in free space. We define the following integral operator: The following Lippmann-Swchinger representation formula holds: Let H j be the generalized eigenspace associated with λ j (ω). Then, from [9], it follows that L 2 (Ω) is the closure of j H j .
Lemma 2.7. Assume that for any j, dim H j = 1, and denote by e j a unitary basis vector for H j . Then the functions form a normal basis for L 2 (Ω × Ω). Moreover,

Pole pencil decomposition of the Green's function
We denote by G(x, y; ω) the Green's function associated with problem (1), that is, the solution in the sense of distributions of satisfying the outgoing radiation condition.
We say that the scattering resonance ω 0 is a non-exceptional scattering resonance if the following assumptions hold: where R(ω 0 ) = 0 and ω → R(ω) is analytic; (ii) The generalized eigenspace H j 0 (ω) is of dimension 1.

Remark 2.9. It is easy to see that for τ large enough, (3) has solutions.
We can now give the following expansion for G when ω is close to a non exceptional scattering resonance. We refer to Appendix A for its proof.

Shift of the scattering resonances by internal small particles
Now let D ⋐ Ω be a small particle of the form D = z + δB, where δ is the characteristic size of D, z is its location, and B is a smooth bounded domain containing the origin. We suppose that D has a magnetic permeability that is different from µ m , and consider the operator where µ = µ c in D and µ = µ m outside D.
As δ → 0, we seek an ω δ in a neighborhood of ω 0 such that there exists a non-trivial solution to subject to the outgoing radiation condition.
The following asymptotic expansion of ω δ holds.
Before proving the above result, we state the following useful lemma. We refer to Appendix B for its proof.
Proof. (of Proposition 3.1) The outgoing solution to problem (5) admits the following Lippmann-Schwinger representation formula: or equivalently, where I denotes the identity operator. Hence, as the characteristic size δ of D goes to zero, we seek ω δ in a neighborhood of ω 0 such that 1/((µ m /µ c ) − 1) is an eigenvalue of T ω δ D . From the pole-pencil decomposition (4) of G, we have Then, it follows that where ( ·, · ) denotes the L 2 real scalar product on D.
Then, (7) can be rewritten as . Now, we need to use the orthogonal decomposition of L 2 (D, R d ) and the spectral analysis of N 0 D on L 2 (D, R d ) that can be found in [13,14]. More precisely, recall that is the space of divergence free L 2 -vector fields and W is the space of gradients of harmonic H 1 functions.
Here, H 1 is the set of function in L 2 having their weak derivatives in L 2 . We will use the following lemma: where ν is the outward normal on ∂D and K * D : L 2 (∂D) → L 2 (∂D) is the Neumann-Poincaré operator associated with ∂D. Moreover, N 0 D | W : W −→ W is a compact operator and hence, the spectrum of N 0 D | W is discrete and the associated eigenfunctions form a basis of W . We refer the reader to [3] for the properties of the Neumann-Poincaré operator. Therefore, using Lemma 3.
see [5] and [6,Lemma 4.2], the term L −1 R[v] can be neglected, and the following asymptotic expansion holds: Moreover, from [6, Proposition 3.1] (see also Appendix C), it follows that where M is the polarization tensor given by [4] M The proof is then complete.

Shift of the scattering resonances by external small particles
Now consider the case where the particle is outside Ω. The main difference is that the modes of K ω Ω are not defined on D, and therefore we must first write the expansion for G outside of Ω. We start by recalling the Lippmann-Schwinger equation for v = G − Γ m : Now, using Proposition 2.10 for z and z ′ inside Ω we have v(z, z ′ ; ω) = c j 0 (ω) e j 0 (z; ω)e j 0 (z ′ ; ω) ω − ω 0 +R(z, z ′ , ω), and we can write an expansion for v(x, The latter equality can be written as where R 1 is regular in space and holomorphic in ω. Let We can now use this expansion in the Lippmann-Schwinger equation again: Therefore, we have an expansion for v outside of Ω: Analogously to the calculations in the previous section, we have for some operator R with smooth kernel that is analytic in ω in a neighborhood V (ω 0 ) of ω 0 . Therefore, by exactly the same method as in the previous section, the following asymptotic expansion can be obtained.

Shift of the scattering resonances due to resonant dispersive particles
Let D ⋐ Ω and suppose that D is made of dispersive material, i.e., such that µ c depends on ω and for a discrete set of frequencies ω, that we can call plasmonic resonances by analogy with the transverse magnetic case, problem (9) (or equivalently the operator µ m + µ c 2(µ m − µ c ) is nearly singular, see [2,7,8]. In that case, we have the following scattering resonance problem: Find ω such that there is a non-trivial solution v to where L(ω) = 1/((µ m /µ c (ω)) − 1)I − N 0 D . Using the Drude model for the permeability, we have where ω p is the volume plasma frequency. It is easy to see that the nearly singular character of (9) is linked to the non-invertibilty of L(ω) on W .
Denote by P 1 : L 2 (D, R d ) −→ L 2 (D, R d ) the orthogonal projector on ∇H 1 0 (D) and P 2 : projector on H(div 0, D). Using Lemma 3.3, we can write the resolvent operator L −1 (ω) as follows: where (λ j , ϕ j ) j are the pairs of eigenvalues and associated orthonormal eigenfunctions of N 0 D . We can then rewrite equation (11) as follows: Now, taking the scalar product on L 2 (D, R d ) with ∇e j 0 and multiplying by (ω − ω 0 )(λ(ω) − λ j ), we obtain that Since R is analytic in ω, the remainder (ω−ω 0 )(λ(ω)−λ j )L −1 R[v] is negligible in a neighborhood of ω 0 . Hence, we arrive at the following proposition: for ω close to ω 0 , then we obtain Hence, we have a significant shift in the scattering resonances if the particle D is resonant near or at the frequency ω 0 . This anomalous effect has been observed in [16].

Asymptotic analysis near exceptional scattering resonances
In this section, we consider the asymptotic behavior of an exceptional scattering resonance for a particular form of the Green's function. These exceptional resonances are due to the non-Hermitian character of the operator T ω D , see [9,15]. For simplicity and in view of the Jordan-type decomposition of the operator T ω D established in [9], we assume that, for ω near ω 0 , G(x, y; ω) behaves like for two functions h (1) and h (2) in L 2 (D). In this simple case, we characterize the shift of the scattering resonance ω 0 due to the small particle D, which is assumed for simplicity to be non-plasmonic. Following the same arguments as those in the previous sections, we seek a non-trivial v such that By multiplying the above equation by ∇h (1) and ∇h (2) , respectively, and integrating by parts over D, we obtain the following system of equations: (v, ∇h (2) ) 1 − c 2 (ω) (L −1 [∇h (2) ], ∇h (2) ) (ω − ω 0 ) 2 = c 1 (ω)(v, ∇h (1) ) (L −1 [∇h (1) ], ∇h (2) ) ω − ω 0 .
Therefore, the following result holds.
Proposition 6.1. Assume that the decomposition (12) holds for ω near ω 0 . Then the perturbed scattering resonance problem (due to the particle D) can be reformulated as a search for ω near ω 0 such that the matrix  is singular.

Concluding remarks
In this paper, the leading-order term in the shifts of scattering resonances of a radiating dielectric cavity due to the presence of small particles is derived. The formula is in terms of the position and the polarization tensor of the particle. It is also proved that the shift is significantly enhanced if the particle is a plasmonic particle and resonates near or at a scattering resonance of the cavity. A characterization of the shift due to small particles near an exceptional scattering resonance is performed. It would be challenging to develop a general theory near such frequencies.
One can check that v(·, x 0 ) is a solution of the following integral equation: Under the assumption that ω 0 is a non exceptional scattering resonance (see Definition 2.8) we can perform a pole pencil decomposition of the resolvent of K ω Ω . We start from the spectral decomposition of the compact operator K ω Ω on L 2 (Ω). The eigenspace associated with the eigenvalue 1 ω 2 0 τ ε c µ m is of dimension one, and we denote by e j 0 its basis. One can then write where ( , ) denotes the L 2 real scalar product on Ω, and ω →R(·, ω) ∈ L 2 (Ω) is analytic in a complex neighborhood V of ω 0 . Using and composing with K ω Ω , we obtain that Using the completeness relation given in Lemma 2.7 yields Γ m (x, y) = j λ j (ω)e j (y)e j (x), for some constantsλ j . Now, we can write that

B Proof of Lemma 3.2
Proof. The operator T D is a singular integral operator of the Calderón-Zygmund type, see [11]. This type of singular operator often arises in electrostatic and magnetostatic theories (see the appendix of [6] for a simple review of the properties of these operators within the formalism of Green's functions) The fact that T ω D is well defined can be deduced directly from Proposition 2.10. Since G can be written as G(x, y) = Γ m (x, y) + K(x, y) where K is a smooth kernel, we can see that the singularity of the derivatives of G is the same as that of the derivatives of Γ m , that is ∂ x i ,x j G(x, y) = ∂ x i ,x j Γ m (x, y) + K i,j (x, y). Therefore, it is easy to see that the singular part of ∂ x i ,x j G(x, y) satisfies the same cancellation property as ∂ x i ,x j Γ m (x, y), that is, Hence, the fact that T D is defined on L 2 (D, R d ) follows directly from classical Calderón-Zygmund theory and the cancellation property above.

C Proof of estimate (8)
Here, we give some more details on how to obtain (8) from the results of [6]. then ∇ϕ solves the integral equation which is exactly Now, replacing ∇e j 0 by its average and controlling the reminder via the Cauchy-Schwartz inequality we have: But the average of ∇ϕ is exactly the dipole moment, which is given by the polarization tensor applied to the average of the exciting field: Since 1 |D|´D ∇e j 0 (x)dx − ∇e j 0 (z) = O(δ) (recall that e j is a mode of the cavity, and is therefore independant of δ) we can replace the average of ∇e j 0 by its value at the center of D to get the result.